Simple Harmonic Motion Quiz

Study Notes

Spring oscillator is a simple harmonic vibration.
The total energy (E) of a spring oscillator is proportional to the product of the mass (m), amplitude (A), and frequency (ω): E = 1/2 mA²ω².

If the mass of a spring oscillator is increased three times and the amplitude is doubled, the total energy becomes 12E.

If a particle participates in two or more simple harmonic motions at the same time, its motion is the synthesis of these harmonic motions.
The natural frequency (ʋ) and natural period (T) of a spring oscillator are related by the equation: ʋ = 2π/T.

The phase (ωt + φ) of a simple harmonic vibration determines the motion state of the vibration system.
The initial phase (φ) is the phase at time zero, and it is related to the displacement (x) and velocity (V) by the equations: x = Acos(ωt + φ) and V = -Aωsin(ωt + φ).

The sine and cosine functions have the following values: sin(0°) = 0, sin(30°) = 1/2, sin(45°) = 1/√2, sin(60°) = √3/2, sin(90°) = 1, cos(0°) = 1, cos(30°) = √3/2, cos(45°) = 1/√2, cos(60°) = 1/2, cos(90°) = 0.

The equation of simple harmonic motion is derived from Newton's second law: F = -kx = ma, where F is the elastic force, k is the spring constant, x is the displacement, and a is the acceleration.
The acceleration of a simple harmonic motion is given by the equation: a = -ω²x, where ω is the angular frequency.
The solution to the motion-differential equation is: x = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the initial phase.